This document discusses dimensional analysis and its applications. It can be used to: 1) Derive equations by ensuring the dimensions on both sides are equal 2) Check if equations are dimensionally correct 3) Find the dimensions/units of derived quantities Examples are provided to illustrate deriving equations based on quantities' dimensions and checking the homogeneity of equations.

1. CHAPTER 1.3: DIMENSIONS CHAPTER 1.3 DIMENSIONAL ANALYSIS

2. Grab the whole picture ! DIMENSIONAL ANALYSIS Measurements Quantities Units Instruments Vector Quantities Scalar Quantities Accuracy & Uncertainty Dimension Analysis Significant Figures

3. DIMENSIONS What is “Dimension” ?

4. Many physical quantities can be expressed in terms of a combination of fundamental dimensions such as [Length] L [Time] T [Mass] M [Current] A [Temperature] θ [Amount] N The symbol [ ] means dimension or stands for dimension

5. There are physical quantities which are dimensionless: numerical value ratio between the same quantity angle some of the known constants like ln, log and etc.

6. Dimensional Analysis Dimension analysis can be used to: Derive an equation. Check whether an equation is dimensionally correct. However, dimensionally correct doesn’t necessarily mean the equation is correct Find out dimension or units of derived quantities.

7. Derived an Equation (Quantities) Example 1 Velocity = displacement / time [velocity] = [displacement] / [time] = L / T = LT -1 v = s / t

8. What are the dimensions of the variables? t -> T m -> M ℓ -> L g -> LT -2 Example 2 The period of a pendulum The period P of a swinging pendulum depends only on the length of the pendulum l and the acceleration of gravity g .

9. T = km a ℓ b g c Write a general equation: T α m a ℓ b g c By using the dimension method, an expression could be derived that relates T, l and g whereby a, b and c are dimensionless constant thus

10. Write out the dimensions of the variables T = M a L b (LT -2 ) c T = M a L b L c T -2c T 1 = M a L b+c T -2c [T] = [m a ][ ℓ b ][g c ] Using indices a = 0 -2c = 1 -> c = -½ b + c = 0 b = -c = ½

11. T = km a ℓ b g c T = km 0 ℓ ½ g - ½ Whereby, the value of k is known by experiment

12. Exercises The viscosity force, F going against the movement of a sphere immersed in a fluid depends on the radius of the sphere, a the speed of the sphere, v and the viscosity of the fluid, η . By using the dimension method, derive an equation that relates F with a, v and η . (given that )

13. To check whether a specific formula or an equation is homogenous Example 1 S = vt [s] = [v] [t] L.H.S [s] = L R.H.S [v] [t] = LT -1 (T) [v] [t] = L Thus, the left hand side = right hand side, rendering the equation as homogenous

14. Example 2 Given that the speed for the wave of a rope is , Check its homogenity by using the dimensional analysis

15. L.H.S [C] = (LT -1 ) 2 [C] = L 2 T -2 R.H.S [F] = MLT -2 , = LT -2 [M] = M Conclusion : The above equation is not homogenous (L.H.S ≠ R.H.S)

16. Exercises Show that the equations below are either homogenous or otherwise v = u + 2as s = ut + ½ at 2

17. Find out dimension or units of derived quantities Example Consider the equation , where m is the mass and T is a time, therefore dimension of k can be describe as

18. -> unit: kgs -2 thus, the units of k is in kgs-2

19. Exercise The speed of a sound wave, v going through an elastic matter depends on the density of the elastic matter, ρ and a constant E given as equation V = E ½ - ρ -½ Determine the dimension for E in its SI units

20. SCALAR AND VECTOR

21. Dimensional Analysis Example: The period P of a swinging pendulum depends only on the length of the pendulum l and the acceleration of gravity g . Which of the following formulas for P could be correct ? (a) (b) (c) Given: d has units of length ( L ) and g has units of ( L / T 2 ) . P = 2  (lg) 2

22. Dimensional Analysis Example continue… Realize that the left hand side P has units of time ( T ) Try the first equation (a) (b) (c) (a) Not Right !!

23. Example continue… Try the second equation Dimensional Analysis (a) (b) (c) (b) Not Right !!

24. Dimensional Analysis (a) (b) (c) (c) This has the correct units!! This must be the answer!! Example continue… Try the third equation